Global well-posedness of 2D Navier–Stokes with Dirichlet boundary fractional noise

Abstract In this paper, we prove the global well-posedness and interior regularity for the 2D Navier–Stokes equations driven by a fractional noise acting as an inhomogeneous Dirichlet-type boundary condition. The model describes a vertical slice of the ocean with a relative motion between the two surfaces and can be thought of as a stochastic variant of the Couette flow. The relative motion of the surfaces is modeled by a Gaussian noise which is colored in space and fractional in time with Hurst parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">H</mml:mi> </mml:mrow> <mml:mo>&gt;</mml:mo> <mml:mfrac> <mml:mn>3</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> .